The synthesis and analysis operators in the g-fusion frames are defined by (for more details, see [1]})
\(T_{\Lambda }&:{H}_2\longrightarrow H \ \ \ \ \ \ \ \ \ , \ \ \ \ \ \ T_{\Lambda }^*:H\longrightarrow {H}_2\\T_{\Lambda }(\lbrace f_j\rbrace _{j\in J})&=\sum _{j\in J}v_j \pi _{W_j}\Lambda _{j}^{*}f_j \ \ \ , \ \ \ \ T_{\Lambda }^*(f)=\lbrace v_j \Lambda _j \pi _{W_j}f\rbrace _{j\in J}.\) 
